Exact calculation of degrees for lattice equations: a singularity approach

Abstract: The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. Here we propose a new method to rigorously compute the exact degree of each iterate for lattice equations. Our strategy is to extend Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (ordinary difference equations) from the singularity structure, to lattice equations. First, we illustrate, without rigorous discussion, how to calculate degrees for lattice equations using the lattice version of Halburd's method and discuss what problems we need to solve to make the method rigorous. We then provide a framework to ensure that all calculations are accurate and rigorous. We also discuss how to detect the singularity structure of a lattice equation. Our method is not only accurate and rigorous but also can easily be applied to a wide range of lattice equations.